## Section 1. Reporting Measurements

The following examples use the metric system for measuring lengths. The ideas are applicable to measurement in other contexts, such as volume or area, and using the English system as well as the metric system. These examples were chosen because it is easier to convey precision with decimals than fractions, and the metric system makes it easy to use decimals. Look at your ruler and notice that the centimeter is divided into tenths (each of those is called a millimeter) and the inch is divided into either eighths or sixteenths. So when we want to make measurements with a ruler smaller than one centimeter, we can easily use decimals, but when we want to make measurements smaller than one inch with a ruler, it is natural to use fractions rather than decimals.

Suppose we are using a ruler to measure the lengths of some pieces of cardboard and we give each measurement, along with a phrase describing how precisely we measured it.

 Measurement 12 cm, correct to the nearest cm. 18.2 cm, correct to the nearest tenth of a cm. 33 cm, correct to the nearest tenth of a cm. 30 cm, correct to the nearest cm. Estimate of 20 cm, correct to the nearest ten cm. Estimate of 180 cm, correct to the nearest ten cm. Estimate of 300 cm, correct to the nearest ten cm. Estimate of 300 cm, correct to the nearest hundred cm.

It would be more convenient to give each number in a way that conveys the precision instead of having to write the phrase afterwards. The generally accepted method for doing that is to report exactly the same number of digits as were observed. That is straightforward in many situations. It is somewhat less straightforward when the most precise digit observed is a “trailing zero.” But that is also easily taken care of for the third number in our list, 33 cm, rounded to the nearest cm. Here we use a trailing zero after the decimal place. In arithmetic courses, we learned that when we compute with the numbers 33 and 33.0, we will obtain the same results. So if someone writes 33.0 instead of 33, they must be intending to convey something more than just the exact value of 33. They are conveying that the number is approximately 33 and also conveying the precision of that approximation.

The fourth number in our list, 30 cm, correct to the nearest cm, can be written as 30. cm. When we explicitly include a decimal we indicate that each of the digits before the decimal was measured precisely. I think of this as only a fairly clear report rather than a clear report, because a decimal point at the end of the number is so easily overlooked or forgotten in copying the number.

 Measurement Completely clear correct report Fairly clear correct report 12 cm, correct to the nearest cm. 12 cm 18.2 cm, correct to the nearest tenth of a cm. 18.2 cm 33 cm, correct to the nearest tenth of a cm. 33.0 cm 30 cm, correct to the nearest cm. 30. cm Estimate of 20 cm, correct to the nearest ten cm. Estimate of 180 cm, correct to the nearest ten cm. Estimate of 300 cm, correct to the nearest ten cm. Estimate of 300 cm, correct to the nearest hundred cm.

The next four numbers in our list are somewhat more difficult to report clearly without words. For a fairly clear correct report, we could say that we will assume that all trailing zeros in the number that would be needed to indicate the size are NOT to be interpreted as conveying precision. If we do that, then three of these four would be interpreted correctly.

 Measurement Completely clear correct report Fairly clear correct report 12 cm, correct to the nearest cm. 12 cm 2.7 cm, correct to the nearest tenth of a cm. 2.7 cm 18.2 cm, correct to the nearest tenth of a cm. 18.2 cm 33 cm, correct to the nearest tenth of a cm. 33.0 cm 30 cm, correct to the nearest cm. 30. cm Estimate of 20 cm, correct to the nearest ten cm. 20 cm Estimate of 180 cm, correct to the nearest ten cm. 180 cm Estimate of 300 cm, correct to the nearest ten cm. Estimate of 300 cm, correct to the nearest hundred cm. 300 cm

We see that there is an advantage in having the most-precise digit in the number after the decimal point when we want to report the number concisely and have the precision be clear. We could do that by changing our scale so that the relevant measurements are all decimals. In this case, since 100 centimeters is 1 meter, we could simply change our measurements to meters from centimeters and then use trailing zeros as needed to convey the precision completely clearly.

The conversion calculation needed for the first measurement in our table is

\begin{align}&12\\\text{cm}\,=\\&=\frac{12\,\text{cm}}{1}\cdot\frac{1\,\,\text{m}}{100\,\text{cm}}\\&=\frac{12\cdot1}{1\cdot100}\,\,\text{m}\\&=\text{0.12}\,\text{m}\end{align}

Review. It is important to be able to convert measurements between different types of units using the proportion method, as illustrated above. Notice that the proportion method enables you to keep track of the units algebraically. This is an important skill. See the course web pages for additional explanations and examples of this method of measurement conversion.

 Measurement Completely clear correct report in centimeters Completely clear correct report in meters 12 cm, correct to the nearest cm. 12 cm 0.12 m 2.7 cm, correct to the nearest tenth of a cm. 2.7 cm 0.027 m 18.2 cm, correct to the nearest tenth of a cm. 18.2 cm 0.182 m 33 cm, correct to the nearest tenth of a cm. 33.0 cm 0.330 m 30 cm, correct to the nearest cm. 0.30 m Estimate of 20 cm, correct to the nearest ten cm. 0.2 m Estimate of 180 cm, correct to the nearest ten cm. 1.8 m Estimate of 300 cm, correct to the nearest ten cm. 3.0 m Estimate of 300 cm, correct to the nearest hundred cm. 3 m